Chi-boundedness conjecture for poset tournaments

A poset tournament is a tournament $T$ for which there exists a total ordering $<$ of $V(T)$ such that, for all $u<v<w$, if $u\to_T v$ and $v\to_T w$, then $u\to_T w$. Equivalently, the forward edges with respect to $<$ form a poset. Poset-tournament chi-boundedness conjecture. The class of poset tournaments is $\vec\chi$-bounded. Poset tournaments were introduced in earlier work, where they were conjectured to characterize tournaments whose $H$-free subclasses have bounded domination number. Whether all poset tournaments are $\vec\chi$-bounded remains open.